NOTES OF MATRICES

POLITEKNIK SULTAN MIZAN ZAINAL ABIDIN

Writic 01 101
02 202
Geometric 03 303
04 404
Notebook

Here is where your
presentation begins

101
202
303
404

101

202 02

04

303

404

101
202
303
404

101
202
303
404

4.0 MATRICES 1

A matrix is 101
a rectangular array 202
303
of numbers 404
enclosed
in large brackets.
For example,




is a matrix.

2

101

01

202 02

03
04

303

404

3

101

01

202 02

03
04

303

404

4

101

01

202 02

03
04

303

404

5

101

01

202 02

03
04

303

404

6

101

01

202 02

03
04

303

404

7

101

202 02

04

303

404

4.1.2 STATE THE 8
TRANSPOSITION OF A
MATRICES 101

01

202 02

03
04

303

404

4.2 OPERATION OF MATRICES 9

02 101
04 202
303
404

4.2.1 ADDITION OF MATRICES 10

Addition of matrices can be carried out by 101
adding the corresponding elements of the
matrices involved. Only matrices of the same
order can be added.

202 02

4.2.2 SUBSTRACTION OF 04
MATRICES

Substraction of matrices can be carried out by 303
substract the corresponding elements of the
matrices involved. Only matrices of the same order
can be substracted

404

4.2.3 MULTIPLICATION OF MATRICES

11

To multiply a matrix with a real 101
number, we multiply

each element with this number

202 02

04

303

404

Multiplication of two matrices, 12

A mxn x B pxq, can only be carried out if n = 101
p ; and the product is a matrix C of order m

x q.

02 202
303
The illustration below shows how two
04
matrices are multiplied:

404

STEPS

1. Multiply each element in the row 13
i- row of matrix A by the
corresponding element in the j- 101
column of matrix B 202
303
2. Sum up all the products 404
obtained in 1. This produces the
elements Cij of matrix C.

3. Repeat steps 1 and 2 until the

elements from all the rows of
matrix A are multiplied by the02

corresponding elements from all

the columns matrix B. 04

If the number TIPS :
of elements in A2 = A X A
row vector is Is not the square of
each element in
the same
as the matrix
number of
rows in the
second matrix
then their
product is not
defined.

EXAMPLE 14

Find the product of each of the following : 101
202
02 303
04 404

15

101

202 02

04

303

404

Activities !!!! 16

101

202 02

04

303

4.

404

4.2.4 DETERMINANT OF MATRICES 17

The determinant of a square matrix is a special number that can 101
be calculated from the matrix. It is used to represent the real-
value of the matrix which can be used to solve simple algebra

problems later on.The symbol for the determinant of
matrix A is det(A) or A.

02 202
303
04 404

b. Determinant of a 3 X 3 matrix

The determinant of a 3x3 matrix is a little more tricky and is
found as follows ( for this case assume A is an arbitrary 3x3
matrix A, where the elements are given below)

EXAMPLE 18

1 101
202
02 303
404
2 04

Activities !!!! 19

101

202 02

04

303

404

4.3 DEMONSTRATE SIMULTANEOUS 20
LINEAR EQUATIONS

101

202 02

04

303

404

4.3.1 INVERSE METHOD 21

STEPS 101
202
02 303
04 404

EXAMPLE 22

02 101
04 202
303
404

23

101

202 02

04

303

404

24

101

202 02

04

303

404

25

101

202 02

04

303

404

4.3.2 CRAMER’S RULE METHOD 26

02 101
04 202
303
404

27

101

202 02

04

303

404

EXAMPLE 28

02 101
04 202
303
404

29

101

202 02

04

303

404

Activities !!!! 30

1. Solve the following 02 101
system of linear 04 202
303
equations using the 404
inverse method

2. Solve the following
system of linear

equations using the
Cramer’s Rule.

3. Solve the following
system of linear

equations using the
inverse method and

Cramer’s Rule.

EXERCISES 31

02 101
04 202
303
404

REFERENCES

32

Ayres Jr, F. & Mendelson, E. (2013). Schaum’s Outlines of Calculus. United States of
America: McGraw-Hill.

Bird, J, (2014). Higher Engineering Mathematics (7th Edition). Glassgow, Routedge.

Fakrul Asraf Daud (2010). Engineering Mathematics 3 For Polytechnics Students 101
- A Problem Solving Approach,Sajadah Ilmu Publication & Distributor, Kuala Lumpur.

Furner, J. and Kumar, D. (2007). The Mathmematics and Science Integration Argument:
A stand for Teacher Education. Eurasia Journal of Mathematics, Science &
Technology Education.

Glyn James. (2012). Advanced Modern Engineering Mathematics.3rd Edition. Pearson 202
Education.

02

H.K Dass (1955). Engineering Mathematics (6th Edition), S.Chand & Company Ltd.,
New Delhi.

04

John Bird (2010). Engineering Mathematics.(6th Edition), Newness

K.A. Stroud & Dexter J. Booth (2011). Advanced Engineering Mathematics (5th Edition), 303
Palgrave Macmillan.

Khoo Cheng, Moy Wah Goon, Tey Kim Soon & Wong Teck Sing (1991). Matematik
Tambahan Tingkatan 4 dan 5, Bersama Enterprise, Selangor.

Sivarama Krishna Das P. and Rukmangadachari E. (2011). Engineering Mathematics.
Volume 1, Second Edition. Pearson Publishing.

Stroud, K. A. & Booth, D. J. (2001). Engineering Mathematics (5th Edition). Great 404
Britain: Palgrave.

Wong Mee Kiong (2014). Soalan-soalan SPM Additional Additional Mathematics
Tingkatan 4 & 5, Sasbadi Sdn. Bhd., Selangor.

BIODATA PENULIS

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